D’ALEMBERT
AND THE DERIVATION
OF THE EQUATION FOR THE
VIBRATING STRING
SM472.5001
CHRISTOPHER M. URBAN
Midshipman First-Class
27 APRIL 2007
Urban.2
Derivation of the Equation for the Vibrating String
For centuries, the motion of a wave fascinated
mathematicians, scientists and engineers. Its motion is as
commonplace as the wind or water; from the echo of music in a
church, sunlight entering through a window, the plucking a
mandolin or violin string, or the simple rolling of the ocean
- the wave and its motion can be found every day.
Mathematically, the equation of a wave was first derived
using a fixed, perfectly elastic vibrating string for a
model. Although different parameters for a vibrating string
will affect the string’s representative equation, the most
basic equations for both a wave and vibrating string are
identical. Consequently, the equation of the vibrating string
may be referred to as the wave equation, and vice versa.
Knowledge of the equation of the vibrating string has
led to the understanding of many phenomena. Water, light and
sound all propagate through waves, and, consequently, the
equations of water, light and sound waves have many
similarities to the equation of a vibrating string. The
understanding and application of the wave equation influences
many topics; optics, seismology, vibration, and sound are all
subjects linked closely to the basic understanding of the
vibrating string.
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Derivation of the Equation for the Vibrating String
It is obvious that an equation representing one of the
most common occurrences in nature can be important. Despite
the equation’s importance, the mathematics needed to derive
such an equation simply did not exist until the 17th century.
Its derivation involved the solution of partial differential
equations, a field not pioneered until the establishment of
calculus in the late 17th century by Leibnitz and Newton.
Partial Differential Equations
A partial differential equation is a relation involving
an unknown function and its partial derivatives. The function
depends on several independent variables (i.e. function f,
depending on x and y); its partial derivative, consequently,
is the derivative of that function with respect to one of
those variables (i.e. derivative of f(x,y) with respect to
x). Partial differential equations are used to solve
processes with a number of independent variables, and can
describe any process that is distributed in space and/or
time. The propagation of sound or heat, fluid flow and
electrostatics are all examples of processes whose solution
involves partial differential equations.
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Derivation of the Equation for the Vibrating String
Jean le Rond D’Alembert
As mathematics caught up to the dilemma of describing
the vibrating string (and more importantly, the motion of a
wave), mathematicians immediately set to work on developing a
solution. In particular, a French scholar of the
Enlightenment took up the task. As an educated thinker who
studied philosophy, physics, engineering and mathematics,
Jean le Rond d’Alembert was an influential and prominent
intellectual of the 17th century. It was d’Alembert who first
published the equation of the vibrating string.
Jean le Rond d’Alembert
D’Alembert contributed much to mathematics. He proposed
that the derivative of a function is the limit of a quotient
of increments, and his work with limits led him eventually to
develop a test for convergence. Known as D’Alembert’s Ratio
Test, it tested series for convergence. Theorem 1: For a
Urban.5
Derivation of the Equation for the Vibrating String
series uk, where k is an integer:
converges if ρ<1. Conversely, the series diverges if
ρ>1.
D’Alembert’s greatest contributions to science are found
elsewhere, however. As chief editor of the Encyclopédie, ou
dictionnaire raisonné des sciences, des arts et des métiers,
or Systematic Dictionary of the Sciences, Arts and Crafts, he
is well known for the encyclopedia’s famous introduction. The
Preliminary Discourse described the purpose of the
encyclopedia: namely, “to change the way people think.”
Carrying immense political importance for its influence in
the French Revolution, it attempted to make knowledge of
philosophy and science available to the common man and to
guide the opinion of its readers. As one of the most famous
contributors and editors, d’Alembert achieved widespread
notoriety for his contributions to the encyclopedia.
Perhaps
d’Alembert’s
most
important
contribution
to
mathematics was his work on the equation of the vibrating
string. D’Alembert was the first to publish the equation in
print;
as
with
most
significant
innovations,
however,
d’Alembert’s publication generated much controversy. Although
he clearly took the lead in deriving an equation for the
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Derivation of the Equation for the Vibrating String
vibrating string, his findings were not conclusive. In the
Traité de l'équilibre et du mouvement des fluides, published
in 1747, d’Alembert uses mathematically simplistic boundary
conditions
to
simplification
arrive
of
at
the
his
solution.
problem
D’Alembert’s
introduces
the
over-
inherent
question of how to handle actual boundary conditions. On top
of this problem, his conclusions could not mathematically
agree with the physical reality of a string’s motion. Both of
these shortcomings naturally brought doubt and discredit to
his solution.
d’Alembert and Euler
About the same time, another influential figure began
work on the derivation of the equation of the vibrating
string. Leonhard Euler, already a well known mathematician
throughout Europe and Russia, gained interest in the results
of d’Alembert’s work on the equation of the vibrating string.
However, upon publishing his work, Euler attributed no credit
to d’Alembert for his groundbreaking work on the subject.
Despite d’Alembert’s objections, Euler refused to cite him as
a contributor. Euler claimed that he was forced to start all
his work from scratch since he could not read d’Alembert’s
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Derivation of the Equation for the Vibrating String
writing. Thus, he did not owe d’Alembert any credit.
This
argument expanded into a great feud between the two
mathematicians, ending only with d’Alembert’s death in 1783,
more than 35 years later.
Wave Equation Derivation
Despite the problems which confronted the d’Alembert
solution, it was the first time the wave equation appeared in
print. The method of d’Alembert to derive the equation is
ingenious, and is worth reproducing. The following proof is
the method d’Alembert derived the wave equation in 1747.
After several attempts, d’Alembert’s application of
differential calculus to the problem of the vibrating string
led him to a partial differentiated solution.
Theorem 2: The partial differential equation:
d2u/dt2 = c2*(d2u/dx2)
is satisfied by the equation
(1)
u = u(x,t) = m(x+t) + n(x-t),
where m and n are arbitrary functions.
Proof: To simplify things, d’Alembert let p = du/dx and let q
= du/dt.
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Derivation of the Equation for the Vibrating String
Since u is a function of x and t,
du = p*dx + q*dt .
(2)
Both partial derivatives of u, one of the x components and
another of the t components, add together to complete the
partial derivative of u. This is half the key to the proof.
The other half of the proof is not obvious, and is the
reason for the proof’s ingenuity. If we rewrite equation (1)
in terms of p and q now, the result is
(3)
The
next
differential
step
known
of
as
d’Alembert’s
an
exact
or
proof
total
involves
differential.
a
A
differential is exact if, for a function f, ∫df is pathindependent. For example, if f = P(x,y)+ Q(x,y), then f is an
exact
differential
generality.
if
Regardless
dp/dx
of
=
the
dq/dy,
variable
without
the
loss
of
function
is
differentiated by, the integral of the function is always the
same.
Using equation (3), we see the possibility of another
equation,
one
equation
(2).
created
in
Realizing
a
fashion
that
p*dt
similar
+
q*dx
to
is
that
an
of
exact
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Derivation of the Equation for the Vibrating String
differential, we can establish a second equation, denoted dv.
This is the most critical piece of the derivation.
dv = p*dt + q*dx
(4)
Reviewing the derivation so far, we have described two
new equations, namely:
du = p*dx + q*dt
dv = p*dt + q*dx
Adding the two equations, one is left with:
du + dv = p*dx + q*dt + p*dt + q*dx = (p+q)*(dx+dt)
(5)
Similarly, after subtracting the two equations:
du – dv = p*dx + q*dt – p*dt – q*dx = (p – q)*(dx – dt))(6)
This result is clearly a partial differential equation,
depending on the independent variables x and t. For u + v,
there
exists
some
function
m
which
depends
on
x
+
t.
Likewise, when u – v is the case, there exists some function
n which depends on x – t.
So, we now have:
and,
u +v = 2*m(x + t)
u – v = 2*n(x – t).
(7)
(8)
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Derivation of the Equation for the Vibrating String
Adding these two identities gives us the result:
2*u = 2*m(x +t) + 2*n(x – t)
(9)
which is simplified to
u = m(x + t) + n(x – t).
Q.E.D.
(10)
Jean le Rond d’Alembert‘s solution explains an important
part of the behavior of a vibrating string or wave. The
function
m
describes
a
left-traveling
wave,
while
the
function n describes a right-traveling wave. The addition of
these two waves produces the resulting visible wave.
Partial Differential Equation Derivation
Theorem 3: The work remaining to be done on the wave
equation is the derivation of equation (1):
d2u/dt2 = c2*(d2u/dx2)
Although
trial
this
and
equation
observation,
was
we
derived
will
by
(11)
d’Alembert
derive
this
differential equation using mathematics and physics.
through
partial
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Derivation of the Equation for the Vibrating String
Proof (by physics):
θ(x+Δx,t )
T(x+Δx, t)
u(x,t)
θ(x,t)
Δu
Δx
T(x,t)
x
Figure 1
Figure 1 shows a segment of a perfectly elastic string,
secured at both ends, in motion. The function u(x,t) is the
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Derivation of the Equation for the Vibrating String
vertical displacement of the string from the x-axis at
position x and time t. The function T(x,t) is the tension in
the string at position x and time t. θ(x,t)is the angle
between the string and a horizontal line at position x and
time t. We will define ρ(x) as the mass density of the string
at position x.
There are three forces acting on the string segment. The
first force is the tension pulling to the right, which has
magnitude T(x+Δx,t) and acts at angle θ(x+Δx,t). The second
force is the tension pulling to the left, which has magnitude
T(x,t) and acts at angle θ(x,t). Finally, all other external
forces, like gravity will be grouped together as a single
force. We shall assume that all of the external forces act
vertically and we shall denote by F(x,t)*Δx as the net
magnitude of the external forces acting on the element of
string.
We begin this proof using Newton’s Second Law. Briefly
stated, Newton’s Second Law states that the force acting on
an object is the object’s mass times its acceleration. By
summing the three forces acting on the string and determining
its mass and acceleration, we can describe the force acting
on the string.
First we will determine the mass of the string. Since
the string element is a tiny segment of the string, there is
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Derivation of the Equation for the Vibrating String
little curvature in the string. Using this assumption, we can
treat the string element as a right triangle, where the
changes in x and u act as the sides of the triangle. Using
the Pythagorean Theorem, the length of the string is thus
√(Δx2 +Δu2). Since density is mass divided by length, the mass
of the element of string is:
ρ(x)*√(Δx2 +Δu2).
(12)
Acceleration is the other undetermined quantity in our
equation. We know that velocity is the distance covered over
time of an object. So, velocity is the time derivative of the
function of an object’s position. Likewise, acceleration is
change
in
derivative
velocity
over
of
position
the
time.
Thus,
function
the
second
is
an
time
object’s
acceleration.
According to Newton’s Second Law, the net force on an
object
is
its
mass
times
acceleration.
So,
adding
the
vertical components of tension and external forces:
ρ(x)*√(Δx2 +Δu2) * d2u/dt2
=
T(x+x,t)*sinθ(x+x,t)-T(x,t)* sinθ(x,t)+ F(x,t)*Δx
(13)
If we divide equation (13) by Δx and take the limit of x
as it approaches 0, we get :
ρ(x)*√(12+(du/dx)2)*d2u/dt2(x,t) =
=d/dx[T(x,t)*sinθ(x,t)] + F(x,t).
(14)
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Derivation of the Equation for the Vibrating String
=dT/dx(x,t)*sinθ(x,t) + T(x,t)*cosθ(x,t)*dθ/dx(x,t)+F(x,t)
√(1+tan2θ)
tanθ
θ
1
Figure 2
Using Figure 2, we can observe that
tanθ(x,t) = limx->0Δu/Δx = du/dx(x,t).
(15)
At this point, our equation is too complicated to solve.
So, since vibrating strings usually have small amplitudes or
vibrations (i.e. guitar string, violin string, etc.) we will
assume that the vibrations in this system are small. This
implies that │θ(x,t)│<<1 for all x and t. Applying this
conclusion, it follows that │tanθ(x,t)│<<1, and from equation
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Derivation of the Equation for the Vibrating String
(15), that │du/dx(x,t)│<<1. Thanks to this conclusion, we
know:
√(12+(du/dx)2)≈1 ,
sinθ(x,t)≈du/dx(x,t),
cosθ(x,t)≈1 ,
dθ/dx(x,t) ≈ d2u/dx2(x,t) (16)
Substituting into equation (14) gives the equation:
ρ(x)* d2u/dt2(x,t) =
dT/dx(x,t)* du/dx(x,t) + T(x,t)* d2u/dx2(x,t) + F(x,t).
(17)
Thanks to our conclusion that │du/dx(x,t)│<<1, we can
simplify this equation further:
ρ(x)* d2u/dt2(x,t) = T(x,t)* d2u/dx2(x,t) + F(x,t). (18)
This settles the vertical component of the string element for
now.
Now we follow similar steps in solving the horizontal
element of the string. Beginning with Newton’s sum of the
forces, T(x+Δx,t)*cosθ(x+Δx,t) – T(x,t)*cosθ(x,t) = 0 (19)
Similarly
dividing
by
Δx
and
taking
its
limit
approaches 0, we have: d/dx[T(x,t)*cosθ(x,t)]=0.
as
Δx
(20)
Next, we make the case that the amplitudes of the string in
our system will be small, or │θ(x,t)│<<1. For a small θ,
vibrations/waves will be directed only horizontally. This
observation implies that tension is a function solely of
time, further simplifying our equation to:
ρ(x)* d2u/dt2(x,t) = T(t)* d2u/dx2(x,t) + F(x,t).
(21)
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Derivation of the Equation for the Vibrating String
Assuming
that
string
density,
ρ,
will
be
constant
throughout the string, and that there are no acting external
forces, F, on the string, we are left with:
d2u/dt2(x,t) = c2* d2u/dx2(x,t) (22)
where c = √(T/ρ).
Q.E.D.
This conclusion, the same made by d’Alembert, is the
partial differential equation that describes the propagation
of a wave over time. Its importance cannot be underestimated.
The equation’s applicability extends far beyond the initial
scope of the vibrating string. Since the days of d’Alembert,
we now know that waves describe the motion of many phenomena;
sound, water and light all travel
in waves, and may be
described using variations of the above 1-D vibrating string
partial differential equation. D’Alembert’s success in 1747
in publishing the solution to the equation of a vibrating
string was the first step in understanding the important
topic of wave propagation.
Urban.17
Derivation of the Equation for the Vibrating String
WORKS CITED
Falstad, Paul. “Loaded String Simulation”
Version 1.5, posted 7/23/05
http://www.falstad.com/loadedstring/
Sharman, R.V. Vibrations and Waves. London:
Butterworths, 1963.
Weisstein, Eric W. "d'Alembert's Solution."
From MathWorld -A Wolfram Web Resource.
http://mathworld.wolfram.com/dAlembertsSolution.html
Wilkins, D.R. “Jean le Rond D’Alembert.”
School of Mathematics: Trinity College, Dublin
http://www.maths.tcd.ie/pub/HistMath/People/DAlembert/Ro
useBall/RB.DAlembert.htmll
Urban.18
Derivation of the Equation for the Vibrating String
O'Connor, J.J. and Robertson, E.F.
School
of Mathematics and Sciences: University of St. Andrews,
Scotland
http://www.history.mcs.standrews.ac.uk/Biographies/D'Ale
mbert.html
Feldman, J. “Derivation of the Wave Equation”
Mathematics Department: University of British Columbia,
Vancouver http://www.math.ubc.ca/~feldman/apps/wave.pdf
The American Mathematical Monthly > Vol. 64, No. 3
(Mar., 1957), pp. 155-157
Stable URL:http://links.jstor.org/sici?sici=0002890%28